Optimal. Leaf size=145 \[ \frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.192641, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {16, 3021, 2748, 2643} \[ \frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3021
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=b^3 \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3}{4} \int \frac{\frac{4 b^2 B}{3}+\frac{1}{3} b^2 (A+4 C) \cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\left (b^2 B\right ) \int \frac{1}{(b \cos (c+d x))^{4/3}} \, dx+\frac{1}{4} (b (A+4 C)) \int \frac{1}{\sqrt [3]{b \cos (c+d x)}} \, dx\\ &=\frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 b B \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}-\frac{3 (A+4 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.237174, size = 123, normalized size = 0.85 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^{2/3} \left (2 \cos (c+d x) \left (C \cos (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )-2 B \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )\right )-A \, _2F_1\left (-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.41, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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